Few-Body Systems

, 60:30 | Cite as

Three-Body Structure of \(^9_{\varLambda }\)Be with \(\alpha \alpha \varLambda \) Cluster Model

  • Jehee Lee
  • Qian Wu
  • Yasuro Funaki
  • Hongshi Zong
  • Emiko HiyamaEmail author
Part of the following topical collections:
  1. Ludwig Faddeev Memorial Issue


In the framework of \(\alpha +\alpha +\varLambda \) three-body cluster model, we calculate energy spectra from bound energy region to resonant energy region up to around 20 MeV with respect to \(\alpha \alpha \varLambda \) threshold. To calculate resonant states, we employ Complex Scaling method which is one of the powerful method. We obtain the states of \(^8\)Be analogue and genuine hypernuclear analogue, which are consistent with those by Yamada et al. (Prog Theor Phys Suppl 81:104, 1985). However, the calculated ordering of \(^9\)Be analogue states is quite different with their calculation. We also obtain \(2^+\) and \(4^+\) resonant states of which have been never pointed out.



The authors would like to thank to Prof. M. Kamimura for useful discussion. This work is supported by RIKEN-MOST project. One of the authors (E. H.) is supported by a Grants-in-Aid for Scientific Research from Monbukagakusho of Japan (18H05407, 16H03995 and 16H02180). One of the authors (Y. F.) is supported by JSPS KAKENHI Grant Number JP18K03658. The numerical calculations were performed at the Yukawa Institute Computer Facility. One of authors (Q.W.) is supported by National Natural Science Foundation of China (11475085, 11535005, 11690030) and National Major state Basic Research and Development of China (2016YFE0129300).


  1. 1.
    T. Motoba, H. Bandō, K. Ikeda, Prog. Theor. Phys. Suppl. 70, 189 (1983)CrossRefGoogle Scholar
  2. 2.
    T. Yamada, T. Motoba, K. Ikeda, H. Bandō, Structure study of typical light hypernuclei. Prog. Theor. Phys. Suppl. 81, 104 (1985)ADSCrossRefGoogle Scholar
  3. 3.
    E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, Y. Yamamoto, Three-body model study of \(A=6 {-} 7\) hypernuclei: halo and skin structures. Phys. Rev. C 53, 2075 (1996)ADSCrossRefGoogle Scholar
  4. 4.
    E. Hiyama, M. Kamimura, K. Miyazaki, T. Motoba, \(\gamma \) transitions in \(A=7\) hypernuclei and a possible derivation of hypernuclear size. Phys. Rev. C 59, 2351 (1999)ADSCrossRefGoogle Scholar
  5. 5.
    B.-N. Lu, E. Hiyama, H. Sagawa, S.-G. Zhou, Superdeformed \(\varLambda \) hypernuclei within relativstic mean field models. Phys. Rev. C 89, 044307 (2014)ADSCrossRefGoogle Scholar
  6. 6.
    M. Isaka, K. Fukukawa, M. Kimura, E. Hiyama, H. Sagawa, Y. Yamamoto, Superdeformed \(\varLambda \) hypernuclei with antisymmetrized molecular dynamics. Phys. Rev. C 89, 024310 (2014)ADSCrossRefGoogle Scholar
  7. 7.
    Y. Funaki, M. Isaka, E. Hiyama, T. Yamada, K. Ikeda, Multi-cluster dynamics in \(^{13}_{\varLambda }\)C and analogy to clustering in \(^{12}\)C. Phys. Lett. B 773, 336 (2017)ADSCrossRefGoogle Scholar
  8. 8.
    K. Tanida et al., Measurement of the \(B(E2)\) of \(^7_{\varLambda }\)Li and shirinkage of the hypernucelar size. Phys. Rev. Lett. 86, 1982 (2001)ADSCrossRefGoogle Scholar
  9. 9.
    R.H. Dalitz, A. Gal, Supersymmetric and strangeness analog states in p-shell \(\varLambda \) hypernuclei. Phys. Rev. Lett. 36, 362 (1976)ADSCrossRefGoogle Scholar
  10. 10.
    R.H. Dalitz, A. Gal, Strangeness analogue states in \(_\varLambda ^9\)Be. Ann. Phys. 131, 314 (1981)ADSCrossRefGoogle Scholar
  11. 11.
    H. Akikawa et al., Hypernuclear fine structure in \(^9_{\varLambda }\)Be. Phys. Rev. Lett. 88, 082501 (2002)ADSCrossRefGoogle Scholar
  12. 12.
    H. Tamura et al., \(\gamma \)-ray spectroscopy in \(\varLambda \) hypernuclei. Nucl. Phys. A 754, 58c (2005)ADSCrossRefGoogle Scholar
  13. 13.
    S. Ajimura et al., Observation of spin–orbit splitting in \(\varLambda \) single-particle states. Phys. Rev. Lett. 86, 4255 (2001)ADSCrossRefGoogle Scholar
  14. 14.
    H. Tamura et al., Observation of a spin-flip M1 transition in \(_\varLambda ^7\)Li. Phys. Rev. Lett. 84, 5963 (2000)ADSCrossRefGoogle Scholar
  15. 15.
    M. Ukai et al., Hypernuclear fine structure in \(^{16}_\varLambda \)O and the \({\varLambda }N\) tensor interaction. Phys. Rev. Lett. 93, 232501 (2004)ADSCrossRefGoogle Scholar
  16. 16.
    M. Ukai et al., Cascade \(\gamma \) decay in the \(_\varLambda ^7\)Li hypernucleus. Phys. Rev. C 73, 012501(R) (2006)ADSCrossRefGoogle Scholar
  17. 17.
    O. Hashimoto, H. Tamura, Spectroscopy of \(\varLambda \) hypernuclei. Prog. Part. Nucl. Phys. 57, 564 (2006)ADSCrossRefGoogle Scholar
  18. 18.
    E. Hiyama, Y. Kino, M. Kamimura, Gaussian expansion method for few-body systems. Prog. Part. Nucl. Phys. 51, 223 (2003)ADSCrossRefGoogle Scholar
  19. 19.
    E. Hiyama, T. Yamada, Structure of light hypernuclei. Prog. Part. Nucl. Phys. 63, 339 (2009)ADSCrossRefGoogle Scholar
  20. 20.
    D.J. Millener, Shell-model interpretation of \(\gamma \)-ray transitions in p-shell hypernuclei. Nucl. Phys. A 804, 84 (2008)ADSCrossRefGoogle Scholar
  21. 21.
    D.J. Millener, Lect. Notes Phys. 724, 31 (2007)ADSCrossRefGoogle Scholar
  22. 22.
    J. Aguilar, J.M. Combes, Commun. Math. Phys. 22, 269 (1971)ADSCrossRefGoogle Scholar
  23. 23.
    E. Balslev, J.M. Combes, Commun. Math. Phys. 22, 280 (1971)ADSCrossRefGoogle Scholar
  24. 24.
    B. Simon, Commun. Math. Phys. 27, 1 (1972)ADSCrossRefGoogle Scholar
  25. 25.
    Y.K. Ho, Phys. Rep. 99, 1 (1983)ADSCrossRefGoogle Scholar
  26. 26.
    N. Moiseyev, Phys. Rep. 302, 211 (1998)ADSCrossRefGoogle Scholar
  27. 27.
    S. Aoyama, T. Myo, K. Kato, K. Ikeda, Prog. Theor. Phys. 116, 1 (2006)ADSCrossRefGoogle Scholar
  28. 28.
    T. Myo, Y. Kikuchi, H. Masui, K. Kato, Prog. Part. Nucl. Phys. 79, 1 (2014)ADSCrossRefGoogle Scholar
  29. 29.
    S. Saito, Effect of Pauli principle in scattering of two clusters. Prog. Theor. Phys. 40, 893 (1968)ADSCrossRefGoogle Scholar
  30. 30.
    S. Saito, Effect of Pauli principle in scattering of two clusters. Prog. Theor. Phys. 41, 705 (1969)ADSCrossRefGoogle Scholar
  31. 31.
    E. Hiyama, M. Kamimura, T. Motoba, T. Yamada, Y. Yamamoto, Three- and four-body cluster models of hypernuclei using the G-matrix \({\varLambda }N\) interaction—\(_\varLambda ^9\)Be, \(_\varLambda ^{13}\)C, \(_{\varLambda \varLambda }^6\)He and \(_{\varLambda \varLambda }^{10}\)Be -. Prog. Theor. Phys. 97, 881 (1997)ADSCrossRefGoogle Scholar
  32. 32.
    E. Hiyama, M. Isaka, M. Kamimura, T. Myo, T. Motoba, Resonant states of the neutron-rich \(\varLambda \) hypernucleus \(^7_{\varLambda }\)He. Phys. Rev. C 91, 054316 (2015)ADSCrossRefGoogle Scholar
  33. 33.
    E. Hiyama, R. Lazauskas, J. Carbonell, M. Kamimura, Possibility of generating a 4-neutron resonance with a \(T=3/2\) isospin 3-neutron force. Phys. Rev. 93, 044004 (2016)ADSMathSciNetGoogle Scholar
  34. 34.
    M. May et al., Observation of hypernuclear gamma-ray transitions in \(_\varLambda ^{7}\)Li and \(_\varLambda ^{9}\)Be. Phys. Rev. Lett. 51, 2085 (1983)ADSCrossRefGoogle Scholar

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© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Physics, H-27Tokyo Institute of TechnologyMeguro, TokyoJapan
  2. 2.Nishina Center for Accelerator-Based ScienceInstitute for Physical and Chemical Research (RIKEN)WakoJapan
  3. 3.Department of PhysicsNanjing UniveristyNanjingChina
  4. 4.Laboratory of PhysicsKanto Gakuin UniversityYokohamaJapan
  5. 5.Joint Center for Particle, Nuclear Physics and CosmologyNanjingChina
  6. 6.State Key Laboratory of Theoretical Physics, Institute of Theoretical PhysicsCASBeijingChina
  7. 7.Department of PhysicsKyushu UniversityFukuokaJapan

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